Uniqueness of positive solutions of fractional boundary value problems with non-homogeneous integral boundary conditions

Fractional Calculus and Applied Analysis

Volumn 15, Issue 3, 2012, Pages 509-528

  Graef, John R ^a   Kong, Lingju ^a   Kong, Qingkai ^b   Wang, Min ^a  

^a UNIVERSITY OF TENNESSEE AT CHATTANOOGA   (United States)

^b NORTHERN ILLINOIS UNIVERSITY   (United States)

Author keywords

Fractional calculus; Integral boundary condition; Non homogeneous boundary condition; Positive solutions; Uniqueness

Indexed keywords

EID: 84869177904     PISSN: 13110454     EISSN: 13142444     Source Type: Journal    
DOI: 10.2478/s13540-012-0036-x     Document Type: Article

References (43)

1. R. Agarwal, D. O'Regan, and S. Stanek, Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations. J. Math. Anal. Appl. 371 (2010), 57-68.
2. A. Aghajani, Y. Jalilian, and J.J. Trujillo, On the existence of solutions of fractional integro-differential equations. Fract. Calc. Appl. Anal. 15, No 1 (2012), 44-69; DOI: 10.2478/s13540-012-0005-4; http://www. springerlink.com/content/1311-0454/15/1/
3. O.P. Agrawal A general solution for the fourth-order fractional diffusion-wave equation. Fract. Calc. Appl. Anal. 3 (2000), 1-12; http://www.math.bas.bg/fcaa
4. H. Bellout, Q. Kong, and M. Wang, Dependence of the unique solution of a periodic boundary value problem on the parameter. Appl. Math. Comput. 217 (2011), 7838-7844.
5. Z. Bai and H. Lü, Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl. 311 (2005), 495-505.
6. A. Cernea, On the existence of solutions for fractional differential inclusions with boundary conditions, Fract. Calc. Appl. Anal. 12, No 4 (2009), 433-442; available at www.math.bas.bg/fcaa/volume12/fcaa124/Cernea fcaa 12 4.pdf
7. A. Cernea, A note on the existence of solutions for some boundary value problems of fractional differential inclusions. Fract. Calc. Appl. Anal. 15, No 2 (2012), 183-194; DOI: 10.2478/s13540-012-0013-4; http://www.springerlink.com/ content/1311-0454/15/2/
8. E. Ait Dads, M. Benchohra, and S. Hamani, Impulsive fractional differential inclusions involving the Caputo fractional derivative. Fract. Calc. Appl. Anal. 12, No 1 (2009), 15-38; available at http://www.math.bas.bg/fcaa/ volume12/fcaa121/Dads Benchohra Hamani FCAA 121.pdf
9. L. Debnath, Fractional integral and fractional differential equations in fluid mechanics. Fract. Calc. Appl. Anal. 6 (2003), 119-156; http://www.math.bas.bg/fcaa
10. A. Dogan, J.R. Graef, and L. Kong, Higher order singular multi-point boundary value problems on time scales, Proc. Edinburgh Math. Soc. 54 (2011), 345-361.
11. L. Erbe, Eigenvalue criteria for existence of positive solutions to nonlinear boundary value problems. Math. Comput. Modelling 32 (2000), 529-539.
12. N.J. Ford and M.L. Morgado, Fractional boundary value problems: analysis and numerical methods. Fract. Calc. Appl. Anal. 14, No 4 (2011), 554-567; DOI: 10.2478/s13540-011-0034-4; http://www.springerlink.com/content/1311-0454/14/4/
13. W. Ge and C. Xue, Some fixed point theorems and existence of positive solutions of two-point boundary-value problems, Nonlinear Anal. 70 (2009), 16-31.
14. C. Goodrich, Existence of a positive solution to a class of fractional differential equations. Appl. Math. Lett. 23 (2010), 1050-1055.
15. J.R. Graef and L. Kong, Existence results for nonlinear periodic boundary value problems. Proc. Edinburgh Math. Soc. 52 (2009), 79-95.
16. J.R. Graef, L. Kong, and Q. Kong, Application of the mixed monotone operator method to fractional boundary value problems. Fractional Differential Calculus, To appear.
17. J.R. Graef, L. Kong, Q. Kong, and M. Wang, Fractional boundary value problems with integral boundary conditions. Submitted.
18. J.R. Graef, L. Kong, and H. Wang, Existence, multiplicity, and dependence on a parameter for a periodic boundary value problem. J. Differential Equations 245 (2008), 1185-1197.
19. J.R. Graef, L. Kong, M. Wang, and B. Yang, Uniqueness and parameter dependence of positive solutions of a discrete fourth order problem. Submitted.
20. J.R. Graef, L. Kong, and B. Yang, Positive solutions for a semipositone fractional boundary value problem with a forcing term. Fract. Calc. Appl. Anal. 15, No 1 (2012), 8-24; DOI: 10.2478/s13540-012-0002-7; http://www.springerlink. com/content/1311-0454/15/1/
21. R. Hilfer, Applications of Fractional Calculus in Physics. World Scientific, Singapore, 2000.
22. D. Jiang, J. Chua, D. O'Regan, and R. Agarwal, Multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces. J. Math. Anal. Appl. 286 (2003), 563-576.
23. D. Jiang and C. Yuan, The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application. Nonlinear Anal. 72 (2010), 710-719.
24. A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Boston, 2006.
25. L. Kong and Q. Kong, Positive solutions of higher-order boundary value problems. Proc. Edinburgh Math. Soc. 48 (2005), 445-464.
26. Q. Kong and M. Wang, Positive solutions of nonlinear fractional boundary value problems with Dirichlet boundary conditions. Electron. J. Qual. Theory Differ. Equ., No. 17 (2012), 1-13.
27. Q. Kong and M. Wang, Eigenvalue approach of even order system periodic boundary value problems. Canad. Math. Bull.; doi:10.4153/CMB-2011-138-3.
28. Q. Kong and M.Wang, Positive solutions of even order periodic boundary value problems. Rocky Mountain J. Math. 41 (2011), 1907-1931.
29. Q. Kong and M.Wang, Positive solutions of even order system periodic boundary value problems. Nonlinear Anal. 72 (2010), 1778-1791.
30. Y. Luchko, F. Mainardi, and S. Rogosin, Professor Rudolf Gorenflo and his contribution to fractional calculus. Fract. Calc. Appl. Anal. 14, No 1 (2011), 3-18; DOI: 10.2478/s13540-011-0002-z; http://www.springerlink.com/content/1311- 0454/14/1/
31. K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations. John Wiley, New York, 1993.
32. K.B. Oldham and J. Spanier, The Fractional Calculus. Academic Press, New York, London, 1974.
33. D. O'Regan and H. Wang, Positive periodic solutions of systems of second order ordinary differential equations. Positivity 10 (2006), 285-298.
34. V. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer-Verlag, New York, 2011.
35. P.J. Torres, Existence of one-signed periodic solutions of some secondorder differential equations via a Krasnosel'skii fixed point theorem. J. Differential Equations 190 (2003), 643-662.
36. I. Podlubny, Fractional Differential Equations. Academic Press, New York, 1999.
37. B. M. Vinagre, I. Podlubny, A. Hern'andez, and V. Feliu, Some approximations of fractional order operators used in control theory and applications. Fract. Calc. Appl. Anal. 3 (2000), 231-248; http://www.math.bas. bg/fcaa
38. G. Wang, D. Baleanu, and L. Zhang, Monotone iterative method for a class of nonlinear fractional differential equations. Fract. Calc. Appl. Anal. 15, No 2 (2012), 244-252; DOI: 10.2478/s13540-012-0018-z; http://www.springerlink.com/ content/1311-0454/15/2/
39. L. Yang and H. Chen, Unique positive solutions for fractional differential equation boundary value problems. Appl. Math. Lett. 23 (2010), 1095-1098.
40. E. Zeidler, Nonlinear Functional Analysis and its Applications I: Fixed-Point Theorems. Springer-Verlag, New York, 1986.
41. C. Zhai and M. Hao, Fixed point theorems for mixed monotone operators with perturbation and applications to fractional differential equation boundary value problems. Nonlinear Anal. 75 (2012), 2542-2551.
42. C. Zhai and L. Zhang, New fixed point theorems for mixed monotone operators and local existence-uniqueness of positive solutions for nonlinear boundary value problems. J. Math. Anal. Appl. 382 (2011), 594-614.
43. S. Zhang, Positive solutions to singular boundary value problem for nonlinear fractional differential equation. Comput. Math. Appl. 59 (2010), 1300-1309.